Notes on the "Ramified" Seiberg-Witten Equations and Invariants

TL;DR

This paper studies the properties of ramified Seiberg-Witten equations with surface operators on four-manifolds, revealing conditions for solutions, invariants behavior, and their dependence on parameters, with some cases showing invariants coincide with ordinary ones.

Contribution

It provides a detailed analysis of ramified Seiberg-Witten equations, identifying admissible surface operators, invariants behavior under different conditions, and their parameter dependencies, extending understanding of these equations.

Findings

Invariants vanish for manifolds with positive scalar curvature and b^+_2 > 1.

Wall-crossings occur when b^+_2 = 1, causing invariants to jump.

In some cases, ramified invariants coincide with ordinary invariants up to a sign.

Abstract

In these notes, we carefully analyze the properties of the "ramified" Seiberg-Witten equations associated with supersymmetric configurations of the Seiberg-Witten abelian gauge theory with surface operators on an oriented closed four-manifold X. We find that in order to have sensible solutions to these equations, only surface operators with certain parameters and embeddings in X, are admissible. In addition, the corresponding "ramified" Seiberg-Witten invariants on X with positive scalar curvature and b^+_2 > 1, vanish, while if X has b^+_2 = 1, there can be wall-crossings whence the invariants will jump. In general, for each of the finite number of basic classes that corresponds to a moduli space of solutions with zero virtual dimension, the perturbed "ramified" Seiberg-Witten invariants on Kahler manifolds will depend - among other parameters associated with the surface operator - on the monopole number "l" and the holonomy parameter "alpha". Nonetheless, the (perturbed) "ramified" and ordinary invariants are found to coincide, albeit up to a sign, in some examples.